How to prove that the problem is a tautology, using only replacement by equivalence(s) (1. negation, 2. distribution, 3. de Morgan's laws, 4. $x\leftrightarrow y\equiv(x\rightarrow y)\wedge(y\rightarrow x)$, 5. $x\rightarrow y\equiv\neg y\rightarrow \neg x$, 6. $\neg(x\rightarrow y)\equiv x\rightarrow \neg y$):
The problem: $(x \wedge(x\rightarrow y))\rightarrow y\\$
What I did was this:
$(x\wedge \neg(x\wedge\neg y))\rightarrow y\\(x\wedge(\neg x\vee y))\rightarrow y\\((x\wedge\neg x)\vee(x\wedge y))\rightarrow y\\$
Now I'm not sure if I'm allowed to do this:
$(F\vee(x\wedge y))\rightarrow y\\$
I conclude nothing changes when I remove $F$, since it's connected with OR operator.
$(x\wedge y)\rightarrow y$
Again, I just conclude it all depends on $y$, and since the values will be the same ($y\rightarrow y)$, it will always be true (tautology).
I understand I didn't actually use only replacements by equivalence, and that's my problem. What can't I see there that can be done to simplify the expression further?
EDIT: ok i got it, i'm an idiot, how to delete this? (there are no delete options)